Varieties via a filtration of the KBSM and knot contact homology

Nagasato, Fumikazu

doi:10.1016/j.topol.2019.06.024

Cited by 3 publications

(9 citation statements)

References 18 publications

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“…This can be understood via Hilbert's Nullstellensatz as follows. (In fact, a similar understanding on this has been discovered earlier in [15].) The coordinate ring C[F 2 (K)] has the following description:…”

Section: 2supporting

confidence: 62%

“…I would like to thank Professor Xiao-Song Lin, who passed away early in 2007, for very fruitful discussions, comments, encouragement, his hospitality during my stay and giving me a lot of opportunities to get important things in my life. The researches at that time was put in [15] as a prototype of this paper. The topics in this paper had been partially supported by JSPS Research Fellowships for Young Scientists, JSPS KAKENHI for Young Scientists (Start-up), and MEXT KAKENHI for Young Scientists (B).…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Some of the results in [19] have already been shown in [18]. This paper also modifies some statements in [18,19] with some progress.This research is motivated by [12,15] to investigate geometric properties of the crosssection of the character variety of a knot with the hyperplanes defined by meridionally trace-free (traceless) conditions, which section we call the trace-free slice of the character variety of a knot (or of a knot, simply). In [18], we have given a set of equations associated with a diagram of a knot whose common solutions coincide with the trace-free slice.…”

mentioning

confidence: 99%

“…We call this set of points the trace-free slice of X(K) (or of K, simply). The trace-free slices have several interesting topological information, for example, a relationship to the knot signature [12], the A-polynomial of a knot [16], the connected sum of knots [17] a 2-fold branched covering structure coming from metabelian representations [13,22], and degree 0 abelian knot contact homology [4,15,19] etc. One of the main results in the present paper is to give concretely a set of equations whose common solutions coincide with the trace-free slice S 0 (K).…”

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confidence: 99%

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Trace-free characters and abelian knot contact homology I

Nagasato

2017

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This paper explains and updates the results [19] given in the talk by the author at the conference "RIMS Seminar, Representation spaces, twisted topological invariants and geometric structures of 3-manifolds" held in 2012. Some of the results in [19] have already been shown in [18]. This paper also modifies some statements in [18,19] with some progress.This research is motivated by [12,15] to investigate geometric properties of the crosssection of the character variety of a knot with the hyperplanes defined by meridionally trace-free (traceless) conditions, which section we call the trace-free slice of the character variety of a knot (or of a knot, simply). In [18], we have given a set of equations associated with a diagram of a knot whose common solutions coincide with the trace-free slice. This gives us a strong tool to describe exactly the relationship between the trace-free slice, degree 0 abelian knot contact homology and the character variety of the 2-fold branched cover of the 3-sphere branched along the knot. Using this, we give a notion called ghost characters of a knot and conditions on them to check(1) when Ng's conjecture concerned with the above relationship does not hold, (2) when the map Φ : S 0 (K) → X(Σ 2 K) constructed [22] is not surjective. Recently, Christopher R. Cornwell [4] translated the results in [19] on the above relationship into the language of "reflective augmentation" by using [18,22].

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Section: 2supporting

confidence: 62%

Section: Acknowledgmentsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Trace-free characters and abelian knot contact homology I

Nagasato

2017

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“…(See Proposition 4.2 in [9]. This was originally found in [7].) So F 2 (K) is an invariant of knots, justifying the notation as F 2 (K).…”

Section: 3mentioning

confidence: 92%

Ghost characters and character varieties of 2-fold branched covers

Nagasato,

Suzuki

2017

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It is known that for any knot K every (meridionally) trace-free SL 2 (C)representation of the knot group G(K) gives an SL 2 (C)-representation of the fundamental group π 1 (Σ 2 K) of the 2-fold branched covering Σ 2 K of the 3-sphere branched along K. In this paper, we show by using a notion called a ghost character of a knot that for the (4, 5)-torus knot T 4,5 the fundamental group π 1 (Σ 2 T 4,5 ) has an SL 2 (C)-representation which cannot be given by any trace-free SL 2 (C)representation of G(T 4,5 ).2010 Mathematics Subject Classification. Primary 57M27; Secondary 57M25. Key words and phrases. character varieties, ghost characters, τ -equivalent representation.1 Such an SL 2 (C)-representation of π 1 (Σ 2 K) is called a τ -equivalent representation. See [10].

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